Contrib Mineral Petrol
DOI 10.1007/s00410-012-0758-0
ORIGINAL PAPER
Multicomponent diffusion in garnets I: general theoretical
considerations and experimental data for Fe–Mg systems
Sascha André Borinski • Ulrich Hoppe •
Sumit Chakraborty • Jibamitra Ganguly
Santanu Kumar Bhowmik
•
Received: 2 November 2011 / Accepted: 12 April 2012
Ó Springer-Verlag 2012
Abstract We have carried out a combined theoretical and
experimental study of multicomponent diffusion in garnets
to address some unresolved issues and to better constrain
the diffusion behavior of Fe and Mg in almandine–pyroperich garnets. We have (1) improved the convolution correction of concentration profiles measured using electron
microprobes, (2) studied the effect of thermodynamic nonideality on diffusion and (3) explored the use of a mathematical error minimization routine (the Nelder-Mead
downhill simplex method) compared to the visual fitting of
concentration profiles used in earlier studies. We conclude
that incorporation of thermodynamic non-ideality alters the
shapes of calculated profiles, resulting in better fits to
measured shapes, but retrieved diffusion coefficients do not
differ from those retrieved using ideal models by more than
a factor of 1.2 for most natural garnet compositions.
Communicated by T. L. Grove.
Electronic supplementary material The online version of this
article (doi:10.1007/s00410-012-0758-0) contains supplementary
material, which is available to authorized users.
S. A. Borinski (&) S. Chakraborty
Institut für Geologie, Mineralogie und Geophysik,
Ruhr Universität Bochum, 44780 Bochum, Germany
e-mail: s.borinski@gmx.de
U. Hoppe
Institut für Computational Engineering, Ruhr Universität
Bochum, 44780 Bochum, Germany
J. Ganguly
Department of Geosciences, University of Arizona,
Tucson, AZ 85721, USA
S. K. Bhowmik
Department of Geology and Geophysics,
Indian Institute of Technology, Kharagpur 721 302, India
Diffusion coefficients retrieved using the two kinds of
models differ only significantly for some unusual Mg–Mn–
Ca-rich garnets. We found that when one of the diffusion
coefficients becomes much faster or slower than the rest, or
when the diffusion couple has a composition that is dominated by one component ([75 %), then profile shapes
become insensitive to one or more tracer diffusion coefficients. Visual fitting and numerical fitting using the NelderMead algorithm give identical results for idealized profile
shapes, but for data with strong analytical noise or asymmetric profile shapes, visual fitting returns values closer to
the known inputs. Finally, we have carried out four additional diffusion couple experiments (25–35 kbar,
1,260–1,400 °C) in a piston-cylinder apparatus using natural pyrope- and almandine-rich garnets. We have combined our results with a reanalysis of the profiles from
Ganguly et al. (1998) using the tools developed in this
work to obtain the following Arrhenius parameters in
D = D0 exp{–[Q1bar ? (P–1)DV?]/RT} for D*Mg and D*Fe:
Mg: Q1bar = 228.3 ± 20.3 kJ/mol, D0 = 2.72 (±4.52) 9
10-10 m2/s, Fe: Q1bar = 226.9 ± 18.6 kJ/mol, D0 = 1.64
(±2.54) 9 10-10 m2/s. DV? values were assumed to be the
same as those obtained by Chakraborty and Ganguly
(1992).
Keywords Convolution effect Garnet Multicomponent
diffusion Numerical model Thermodynamic non-ideality
Introduction
Compositionally zoned garnets occur in a wide variety of
geological settings that include hydrothermal systems,
skarns, acidic volcanics, igneous granites and their contact metamorphic aureoles, regional metamorphic pelites,
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Contrib Mineral Petrol
calc-silicates and metabasites, and ultramafic rocks from
the mantle (see, e.g., Chakraborty and Ganguly 1991; Kohn
2003, for reviews). The preservation and modification of
compositional zoning has been used widely to obtain
timescales of various processes (e.g., cooling rates, exhumation rates) and as records of tectonic-, reaction-, deformation- and fluid-flow histories. A key parameter in such
analyses is the knowledge of the relevant diffusion coefficients in garnets. A significant body of diffusion coefficients for different species in various garnet compositions
is now available, and these have been recently reviewed by
Ganguly (2010).
Most experimental studies have focused on almandine–
pyrope-rich garnet solid solutions that occur commonly in
metamorphic rocks or in the mantle. Given the multicomponent nature of diffusion in garnet and the unavailability of
suitable crystals to cover the compositional space adequately (e.g., nine crystals of well-defined composition to
extract a 3 9 3 multicomponent diffusion matrix), the
approach pioneered by Loomis et al. (1985) has been used by
most authors (e.g., Chakraborty and Ganguly 1992; Ganguly
et al. 1998; Perchuk et al. 2009) to determine diffusion
coefficients from multicomponent exchange experiments
with garnets. This approach makes use of the relationship
between the different elements of the D-matrix via tracer
diffusion coefficients. Measured concentration profiles
generated by chemical diffusion (diffusion in response to a
chemical concentration/chemical potential gradient) are
fit to retrieve four tracer diffusion coefficients (D*Fe, D*Mg,
D*Mn and D*Ca). We will refer to this approach as ‘‘multicomponent modeling’’ in the following. For applications,
these tracer diffusion coefficients are then recombined using
the same relationship to obtain D-matrices for various garnet
compositions. The use of effective binary diffusion coefficients (EBDC) to model the multicomponent diffusion
profiles (e.g., Chakraborty and Ganguly 1992; Vielzeuf et al.
2007) also relies implicitly on such connections. The manner
in which the approach of Loomis et al. (1985) has been used
so far relies, however, on several assumptions:
(1)
(2)
(3)
The model relating chemical diffusion coefficients
(the elements Dij of the D-matrix in a multicomponent system) to tracer diffusion coefficients (D*i )
developed by Lasaga (1979) is valid.
The effect of thermodynamic non-ideality has been
ignored in all analyses so far, and the garnets have
been assumed to be ideal solutions.
A special case of the convolution correction of
Ganguly et al. (1988) for systems with a single,
constant diffusion coefficient was applicable to the
measured concentration profiles in the multicomponent diffusion experiments using pyrope-almandine
couple.
123
(4)
The visual fitting of the concentration profiles to
retrieve tracer diffusion coefficients provides true best
fits in a statistical sense and is unique.
Moreover, there remain issues as to the extent to which
crystals used for the measurement of diffusion coefficients
in the laboratory are representative of compositions found
in the geological settings described above (e.g., Carlson
2002, 2006), and the parameters describing the diffusion
coefficient of Ca remain poorly defined (Ganguly 2010).
The first assumption was tested partially by Chakraborty
and Rubie (1996) when they measured Mg tracer diffusion
coefficients directly and found that the results agreed
within error with extrapolation of data retrieved from
multicomponent modeling. We have carried out new
experiments and theoretical analysis to explore the validity
of assumptions (2)–(4). We consider the experimental
results and their implications for the diffusion of Fe and
Mg in this paper; the diffusion behavior of Ca, which is
found to be affected by added complications, is treated in a
companion paper (referred henceforth as Part-II).
The paper is organized as follows. First, we discuss the
compositional space spanned by experimental data and
how these relate to compositional dependence of diffusion
coefficients and occurrences in natural systems. Next, we
present the theoretical background for handling of experimental data that are developed in this paper. This section
covers three aspects: a model for relating tracer diffusion
coefficients to elements of the D-matrix in a 4-component
non-ideal garnet solid solution, convolution correction of
an arbitrary profile shape (i.e., without the assumption of
single, constant diffusivity) and a statistical procedure for
evaluating the fit between calculated and experimentally
measured profiles in order to retrieve optimized values for
the tracer diffusion coefficients. This is followed by a
discussion of some general behavior of the diffusion profiles in this system and the sensitivity of profile shapes to
values of different tracer diffusion coefficients. After this,
the experimental methods and data are presented. The data
from this study and other related works are then considered
in order to retrieve Arrhenius parameters, and some
implications of these results for applications to natural
systems (e.g., closure temperatures) are discussed to conclude the paper.
Compositional space of garnets in experiments
versus nature
Before embarking on a theoretical analysis of multicomponent diffusion behavior in garnets, it is worth evaluating
to what extent the compositions of the garnet crystals used
for laboratory measurements correspond to those in which
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compositional profiles is measured and interpreted in
natural systems. Much of the discussion of compositional
zoning of garnets focuses on the dodecahedral cations,
and therefore, (Fe, Mg, Mn, Ca)3Al2Si3O12 is a suitable
space to consider for this purpose. Considering ratios or
the use of terms such as ‘‘almandine-rich’’ is ambiguous
in that the concentration of the remaining dodecahedral
cations remains inadequately specified in this description.
To address this issue, we have plotted compositions from
a number of metamorphic and mantle-derived garnets in
a Fe–Mg–Mn–Ca tetrahedron, and in Ca–Mg–Fe and
Mn–Mg–Fe ternaries and overlain the composition of
diffusion couples used in laboratory experiments on these
(Fig. 1).
Several aspects emerge from this comparison: (1) A
vast majority of metamorphic garnets are nearly binary
almandine–pyrope solid solutions that are relatively poor in
the spessartine and the grossularite components. Many
experimental diffusion couples span this range exactly. (2)
Low-grade garnets are often rich in the spessartine component. The compositional range of these garnets is well
covered by the experimental diffusion couples of Chakraborty and Ganguly (1992). (3) Compositions of some
mantle-derived and granulitic (particularly mafic granulites) garnets that are somewhat rich in the grossularite
component are not well described by available experimental diffusion couples. However, most of these compositions are not very far removed from the compositions of
many experimental diffusion couples. (4) Garnets from
eclogitic rocks can be quite rich in the grossularite component, and these are not represented by any experimental
diffusion couples. The implications of this comparison are
that (a) the diffusion of low-grade, Mn-rich garnets occurs
along different compositional vectors compared to that in
most other, Fe–Mg-rich garnets. In the rest of this paper,
we will refer to diffusion along these two compositional
vectors as diffusion in ‘‘Mn-rich’’ and ‘‘Fe–Mg’’ garnets,
respectively. It is implied that in all cases multicomponent
diffusion of all four cations (Fe, Mg, Mn and Ca) occurs;
the designation is merely an indication of the dominant
compositional vector. (b) With the exception of certain
eclogites, compositions of most natural metamorphic and
mantle-derived garnets correspond very well to the range
covered by experimental diffusion couples. Consideration
of the diffusion couples along the ‘‘Fe–Mg’’ vectors also
reveals that the compositional range spanned by a given
couple can be quite variable. In some cases, the compositions of two ends of the diffusion couples are fairly close to
each other so that the compositional vectors are short.
Within this limited compositional range, it is possible to
treat a diffusion coefficient as a constant and compositional
profiles in such couples can be modeled adequately with a
constant diffusion coefficient. On the other hand, there
are couples that span almost the entire range of Fe–Mg
compositions. For these couples, it is essential to consider
the compositional dependence of diffusivity in modeling
the compositional profiles (they often have a pronounced
asymmetry in shape, e.g., as a result of such compositional
dependence of diffusivity). This aspect will become
important in the discussion that follows.
Diffusion in ‘‘Mn-rich’’ garnet was studied by Chakraborty and Ganguly (1992) and Perchuk et al. (2009) only
[an earlier study by Elphick et al. (1985) in the same laboratory used similar garnets, but those data were considered together in Chakraborty and Ganguly (1992)]. Most
other diffusion experiments with multicomponent garnets
dealt with ‘‘Fe–Mg’’ garnets. It is found that there is a
qualitative difference between diffusion along these two
kinds of compositional vectors/gradients. For example, in
Mn-rich garnets, D*Fe*D*Mg, whereas in ‘‘Fe–Mg’’ garnets,
D*Fe \ D*Mg (e.g., see Ganguly et al. 1998). Therefore, it is
best to handle diffusion in these two systems separately. In
this paper, we will focus on diffusion in ‘‘Fe–Mg’’ systems.
Theoretical developments
A multicomponent diffusion model for non-ideal garnet
solid solutions
According to the Lasaga (1979) model, the tracer diffusion
coefficients and the elements, Dij, of the [D] matrix are
related to each other by
D zi zj Xi Dj Dn
Dij ¼ Di dij Pn i 2
k¼1 Dk zk Xk
ð1Þ
in an ideal system and by
.Xn
o ln ci o ln ci zj
Dij ¼ Di dij þ Di Xi
D z2 X
Di zi Xi
k¼1 k k k
oXj
oXn zn
"
#
n
X
o ln ck zi
o ln ck
zj D j D n þ
X k z k Dk
oXj
oXn zn
k¼1
ð2Þ
in a non-ideal system, where dij = 0 if i = j and 1 if
i = j. D*i , Xi, zi and ci represent the self-diffusion coefficient, molar fraction, valence and activity coefficient of
component i, respectively.
To use Eq. (2), it is necessary to have an activitycomposition model that allows one to calculate the (qlnci/
qXj) terms. For the garnet compositions of interest, such
activity-composition relations have been determined
experimentally by Ganguly et al. (1996), who treated the
garnet solid solution according to a multicomponent solution model that was developed by Cheng and Ganguly
(1994), which is based on a Taylor series expansion of
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Contrib Mineral Petrol
the multicomponent Gibbs free energy surface, as in the
ternary formulation of Wohl (1953). This formulation has
the attractive property that the compositions of the
bounding binaries that are used to calculate the multicomponent free energy turn out, as a mathematical consequence, to be those obtained by the normal projections of
the multicomponent composition on to the binaries. The
normal projection method was suggested by the widely
used Redlich–Kister–Muggianu formulation (Muggianu
et al. 1975). Ignoring the ternary interactions, the general
expression of ln ci in a multicomponent solution with
subregular binaries is given, according to Cheng and
Ganguly model, by
"
n
X
1
ln ci ¼
ð1 2Xi Þ
Wij Xj2 þ 2Xi ð1 Xi Þ
RT
j¼16¼i
n
X
Wji Xj 2
j¼16¼i
þ
n
n
X
X
Wjk Xj Xk2
j¼16¼i k¼16¼i;j
n1
X
n
X
ð1 2Xi Þ
Xj
Xk ðWij þ Wik þ Wji
2
j¼16¼i
k¼jþ16¼i
þ Wjk þ Wki þ Wkj Þ n2
X
Xj
j¼16¼i
n
X
n1
X
Xk
k¼jþ16¼i
#
Xl ðWjk þ Wjl þ Wkj þ Wkl þ Wlj þ Wlk Þ
l¼kþ16¼i
ð3Þ
where Wij represents a subregular interaction parameter in
the binary join i - j. The interaction parameters are
independent of composition, but depend on pressure and
temperature, and can be expressed as (Ganguly and Saxena
1987)
Wij ðP; TÞ ¼ WijH ð1; TÞ TWijS ð1; TÞ þ ðP 1ÞWijV ;
j¼16¼i;m
Wjm Xj n1
X
j¼16¼i;m
Xj
#
þWjm þ Wjk þ Wkm þ Wkj Þ
123
n
X
"
o ln ci
1
¼
2ð1 2Xi ÞWim Xm þ 2Xi ð1 Xi ÞWmi
RT
oXm
n
X
2
ð2Wjm Xj Xm þ Wmj Xj2 Þ
j¼16¼i;m
þ
n
ð1 2Xi Þ X
Xj ðWim þ Wij þ Wmi þ Wmj þ Wji þ Wjm Þ
2
j¼16¼i;m
n1
X
j¼16¼i;m
Xj
n
X
#
Xk ðWmj þ Wmk þ Wjm þ Wjk þ Wkm þ Wkj Þ
k¼jþ16¼i;m
ð5bÞ
for all other cases.
Figure 2 shows a set of compositional profiles calculated using ideal solution model, as in the earlier studies
(Chakraborty and Ganguly 1992; Ganguly et al. 1998;
Perchuk et al. 2009; Carlson 2006) those calculated using
the same set of diffusion parameters, but with a non-ideal
model using the interaction parameters in Table 1. The
diffusion parameters and times for which the profiles are
calculated correspond to parameters and experimental run
durations that will be encountered below. Various pairs
were chosen to study the effects and four extreme situations: (a) a diffusion couple close to the Fe–Mg join representing a composition from one of our experimental runs,
(b) a diffusion couple lying close to the Fe–Ca join that
might be found in an amphibole facies rock, (c) a diffusion
couple where a high Fe- and a high Mn-rich composition,
both including substantial Mg- and Ca-bearing components, are coupled—these represent compositions from
certain granulitic rocks and cover a region of compositional
space where some of the most non-ideal garnets may occur,
and (d) a diffusion couple that spans the Mg-rich area of
the tetrahedral space, but including substantial Mn- and
Ca-bearing components, which stands for the composition
ð4Þ
S
V
where WH
ij , Wij and Wij are, respectively, the enthalpic,
entropic and volumetric components. In aluminosilicate
garnets, values of these components are taken from Ganguly et al. (1996) and are reproduced here for convenience
(Table 1).
Using Eq. (3), (qlnci/qXj) terms required to calculate Dij
according to Eq. (2) takes the form
"
n
X
o ln ci
1
2
¼
Wmj Xj2 þ 2ð1 2Xm Þ
RT
oXm
j¼16¼i;m
n
X
when i = m and
Xk ðWmj þ Wmk
k¼jþ16¼i;m
ð5aÞ
Table 1 Summary of the internally consistent binary interaction
parameters in aluminosilicate garnets after Table 4 in Ganguly et al.
(1996)
Parameter (ij)
WH
ij
(J mol-1)
WSij
(J mol-1 K-1)
WVij
(J mol-1 bar-1)
CaMg
21,627
5.78
0.012
MgCa
9,834
5.78
0.058
CaFe
873
1.69
0
FeCa
6,773
1.69
0.03
MgFe
2,117
0
0.07
FeMg
695
0
0
7.67
0.04
MgMn
12,083
MnMg
12,083
7.67
0.03
FeMn
539
0
0.04
MnFe
539
0
0.01
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Fig. 1 Fe–Mg–Mn–Ca tetrahedron and ternary Fe–Mg–Ca and Fe– c
Mg–Mn projections show typical compositions of garnets in garnetbearing rocks (triangles in tetrahedron, fields in ternary diagrams)
and compositions of garnets used for experimental diffusion studies
(diamonds). Each connecting line between the experimental sample
compositions (diamonds) corresponds to one diffusion couple.
Ternary projections additionally show diffusion couples used in
Fig. 2 as circles with connecting lines. These lines represent
hypothetical diffusion couples used for illustrating the effects of
thermodynamic non-ideality on shapes of diffusion profiles in garnet
(shown in Fig. 2). Data used for drawing this figure are available
online as electronic supplementary material
of garnets in certain mantle xenoliths (see Fig. 1). In all of
the four situations, compositions with a relatively high
concentration of Ca were considered, that is, where a
strong non-ideality effect is expected. In addition, in the
last two cases, high concentrations of (Mn and Mg) were
also present on at least one side of the diffusion couple.
This is an unusual situation, but one where effects of nonideality are expected to be maximized because of the high
values of the Mg–Mn interaction parameter (Table 1). The
difference between the profile shapes calculated using the
ideal and the non-ideal formalisms is found to be minimal
in the first three cases (*1 mol. % at any point along the
profile). This is consistent with the expectation based on a
quasi-binary approximation of the mixing property (Chakraborty and Ganguly 1992). The diffusion coefficients that
would best describe a given profile using either the ideal or
the non-ideal model lie within factor of 1.2 of each other.
An exception is given in the fourth case where the Mg, Ca
and Mn concentrations are higher. Here, the difference
between the shapes obtained using the two formalisms may
result in differences of up to *4 mol. % at some spatial
coordinates, and the diffusion coefficients lie within a
factor of 2.5. Such a difference is significant and is an
example where non-ideality cannot be neglected in these
compositionally unusual situations, since only a rare variety of mantle-derived garnet have relatively high Mn
content (6 mol. %) in addition to high Mg and Ca. This is a
case where the effect of non-ideality is maximized because
of strong Mg–Mn and Mg–Ca interactions. In all cases, the
diffusion coefficients required to describe a given profile
using a non-ideal model will be greater. What is noticeable,
however, is that although profile lengths are not affected
substantially, the profile shapes are modified and some
inflections/changes of curvature (including effects of uphill
diffusion) that are obtained with a non-ideal model cannot
be produced using an ideal model (Fig. 2). Moreover, the
effects of non-ideality are different for different elements,
and the symmetry of profile shapes changes when non-ideal
models are considered. For example, the point of inflection
of a profile can be shifted from the central segment
around the interface of the diffusion couple, creating the
123
Contrib Mineral Petrol
a1
a2
b1
Mg
Fe
0.70
0.03
b2
0.03
0.70
Mn
Ca
-0.01
0.30
0.20
0.60
Ca
Mg
0.50
0.01
0.40
Mn
-0.01
0.30
Mn
0.20
0.10
-1
0
1
2
-2
2
0
-2
-1
Distance [µm]
c1
c2
Ca
0.01
0.40
Mn
-0.01
Fe
0.20
N o r m . C o n c e n tr a tio n
Mg
0.50
Cnon-ideal - C -ideal
Mn
Mg
2
-2
Mg
Mg
0.03
0.60
Fe
0.50
0.01
0.40
Fe
Mn
-0.01
0.30
0.20
Ca
Ca
0.10
-0.03
-2
-1
0
1
2
2
0
d2
0.70
0.60
Ca
1
d1
0.03
0.30
0
Distance [µm]
Fe
0.70
-0.03
-2
0
2
Distance [µm]
0.10
-0.03
Mn
-2
-1
Cnon-ideal - C -ideal
-2
Mg
0.10
-0.03
Mn
Ca
N o r m . C o n c e n t r a t io n
Fe
Ca
Cnon-ideal - C -ideal
0.40
0.01
N o r m . C o n c e n tr a tio n
Mg
Fe
0.50
Cnon-ideal - C -ideal
N o r m . C o n c e n tr a tio n
Fe
0.60
0
1
2
-2
0
2
Distance [µm]
Fig. 2 a1–d1 Four sets of simulated diffusion profiles with arbitrarily
chosen initial composition differences. All profiles have been
simulated with the thermodynamically ideal (dashed lines) and nonideal solution (solid lines) models using a constant D-matrix. All
profiles are calculated using the following set of diffusion coefficients
(Mg 1.03 9 10-19, Fe 4.00 9 10-20, Mn 3.79 9 10-19, Ca
2.46 9 10-20 m2/s). These diffusion coefficients are representative
for 28 kbar, 1,050 °C. Profile lengths correspond to an annealing time
of 440 h. Note that D*-values and run conditions used here are
equivalent to the experimental run DPA4. Vertical dashed lines
represent initial point of contact between garnets of the diffusion
couple. a2–d2 Difference between concentration profiles calculated
using ideal and non-ideal solution models, respectively
appearance of a Kirkendall shift such as those known in
metallic systems (Smigelskas and Kirkendall 1947).
Therefore, statistical measures of quality of fit are likely to
be improved when a non-ideal model is used, and we have
used this model (Eq. 2) for analyzing the data discussed in
this paper.
1998), it is necessary to correct for the convolution that
arises from the spatial averaging effect of the microprobe
beam. Ganguly et al. (1988) developed an analytical formulation to address this problem, assuming that the excitation volume for a spot analysis has Gaussian distribution
of intensity with radial symmetry about the axis of the
electron beam. They also developed an analytical solution
in which the diffusion coefficient determined from a
measured diffusion profile, which conforms to the solution
of the diffusion equation with constant diffusion coefficient, is directly corrected for the convolution effect in the
microprobe measurement of the profile. The second
method was applied by Ganguly et al. (1998) to deconvolve
Convolution analysis
It has been shown that for the slow diffusion rates and
resulting short profile lengths (often \10 lm) encountered
in experimental studies of diffusion in garnet (e.g., Elphick
et al. 1985; Chakraborty and Ganguly 1992; Ganguly et al.
123
Contrib Mineral Petrol
a
0.70
Fe
0.60
N o rm . C o n c .
the tracer diffusion coefficients retrieved from modeling
multicomponent diffusion profiles in the pyrope–almandine
couples. However, this procedure is not entirely satisfactory because the tracer diffusion coefficient of a species is
not solely responsible for its diffusion profile in a multicomponent system. Hence, it is desirable to adopt the more
general deconvolution analysis. Since we use a numerical
finite difference scheme to calculate diffusion profiles, it is
best to simply evaluate the convolution integral directly
without any further approximations. This is done using
Ca
0.50
0.40
0.30
Mn
0.20
Mg
0.10
fi ðxÞ ¼ðue Xi ÞðxÞ
Zþ1
¼
ue ðn xÞXi ðnÞdn
-3
ð6Þ
-2
-1
0
1
2
3
Distance [µm]
1
in which e represents the error standard deviation of the
Gaussian or convolution factor. In our simulations of
experimental diffusion profiles, we calculate profiles of all
four components with an assumed set of (tracer) diffusion
coefficients for the given run duration and then convolve
the profiles using Eq. (6). This convolved profile is then
compared to the experimental profiles determined by
sequential spot analyses along a line traverse in an electron
microprobe. The convolution factor (e = 0.48 lm) was
determined earlier by Ganguly et al. (1998) by comparing
profiles across a natural garnet–garnet diffusion couple
measured by electron microprobe and analytical TEM,
assuming that the convolution effect in the latter was
negligible.
Objective fits to experimental diffusion profiles
One of the concerns, in spite of the repeated use of the
visual fitting technique introduced by Loomis et al. (1985),
is that an objective measure of the quality of fit has not
been used in the method. Further, it has remained open to
question how unique the retrieved diffusion coefficients are
(can a given profile be described equally well by two different sets of diffusion coefficients?) and how sensitive the
profile shapes are to small perturbations in the values of the
diffusion coefficients. We have now developed a method to
objectively test these aspects. The method is based on the
Nelder-Mead algorithm or simplex search algorithm
(Nelder and Mead 1965), which is one of the best-known
algorithms for multidimensional unconstrained optimization without derivatives. The method is widely used to
b
0.70
0.60
N o rm . C o n c .
where fi(x) is the observed concentration and Xi(x) is the
true concentration of component i at a given point of the
diffusion profile. ue defines the so-called standard normal
(or Gaussian) density function
1
1 x2
uðxÞ ¼ pffiffiffiffiffiffi exp ð7Þ
2 e
e 2p
Fe
Ca
0.50
0.40
0.30
Mn
0.20
Mg
0.10
-3
-2
-1
0
1
2
3
Distance [µm]
Fig. 3 Two sets of simulated diffusion profiles using the NelderMead optimization algorithm with an arbitrarily chosen initial
composition. The profiles have been calculated with the same
diffusion coefficients and simulation conditions as given in Fig. 2.
a In the first set of simulations, the calculated diffusion profiles have
been fitted to obtain diffusion coefficients that agree with the true
values within three decimal places and the best-fit profiles are
indistinguishable from the input (i.e., what would be measured)
profiles. b Diffusion coefficients retrieved in the second set of
simulations, where a bit of random noise was added to the profiles,
have a small, but measurable deviation from the real diffusion
coefficients even though the best fit and input (i.e., measured) profiles
match very well. Diffusion coefficients differ by a factor of 1.01 for
Mg, 1.09 for Fe, 0.98 for Ca and 1.03 for Mn
solve parameter estimation and similar statistical problems,
where the function values are uncertain or subject to noise.
By progressively moving away from the poorest value, it
locates the minimum in the difference between an objective
function and measured values in a many dimensional space
(e.g., the space of the four tracer diffusion coefficients, in
our case). The validity of the method is demonstrated
(Fig. 3) by calculating diffusion profiles with a given set of
diffusion coefficients and then using arbitrary starting seed
values to test whether the ‘‘true’’ diffusion coefficients are
retrieved by the optimization program. Profiles calculated
123
Contrib Mineral Petrol
according to the retrieved ‘‘best-fit’’ diffusion coefficients
are shown for comparison in Fig. 3. In a second set of
simulations, random noise was added to the calculated
profiles to simulate analytical error and other sources of
uncertainty in the measurement of compositions. For a
range of values, agreements between the retrieved and true
diffusion coefficients are excellent, with a maximum
recorded difference of a factor of 1.09. One of the issues
that have remained unaddressed explicitly is, to what
extent is it possible to retrieve four tracer diffusion coefficients from one set of diffusion profiles? The optimization
algorithm tests this aspect as well and demonstrates that,
depending on the region of compositional space, it is
possible to retrieve four tracer diffusion coefficients from a
single diffusion couple experiment. We explore this aspect
further below.
We note here that in the absence of a known function
that defines the compositional dependence of tracer diffusion coefficients, it is not possible to determine best-fit
diffusion coefficients from compositional profiles that are
visibly strongly asymmetric, requiring compositionally
dependent tracer diffusion coefficients to describe their
shapes. We have used the algorithm to retrieve diffusion
coefficients from experimental diffusion couples where the
compositional range spanned by a diffusion couple is not
large (e.g., the red compositional vectors in Fig. 1) and the
measured profiles are reasonably symmetric. This is discussed below after we present the results from the new
diffusion experiments.
Model calculations to test sensitivity and robustness
To explore how sensitive the profile shapes are to the
values of diffusion coefficients and how uniquely the
shapes of these profiles are related to the different tracer
diffusion coefficients, we have carried out a number of
numerical simulations. In our exploration of compositional
space, we have restricted ourselves to the range found to be
relevant for the garnet compositions commonly found in
nature (Fig. 1). The overall nature of the two effects can be
seen by inspecting the equation relating binary inter-diffusion in an ideal, ionic system of equally charged species
(Lasaga 1979):
Dij ¼
Di Dj
Di Xi þ Dj ð1 Xi Þ
ð8Þ
when all D*i s are similar to each other (as, e.g., is likely for
the diffusion of similarly charged cations on a given lattice
site of a particular mineral), then as the content of one
component becomes very dilute, Dij*D*i . The magnitude
of D*j plays no significant role in determining the profile
123
shapes, and conversely, D*j cannot be determined by fitting
measured profiles. For multicomponent garnets with compositions relevant for experimental and natural samples, we
have carried out a series of simulations to explore the
compositional limit where this effect appears. It is found
that when the concentration of any one component exceeds
75 mol. %, profile shapes of all components begin to get
insensitive to the tracer diffusivity of that element, that is,
the tracer diffusivity of that element can be poorly constrained from measured profile shapes. An example for four
components is shown in Fig. 4. This suggests that diffusion
couples made of intermediate garnet compositions yield the
most well-constrained diffusion data.
The second effect arises when one of the D*i s values
differs significantly from that of all others. This effect can
be explored using the binary equation (Eq. 8) as well.
When D*i D*j , then Dij*D*j /Xi. Simulations in the multicomponent systems show that when the D* value for any
one element is more than four orders of magnitude faster
than that of all others, the profile shapes become insensitive
to further changes in D* of this element. For example, the
profile shapes for all elements are practically identical
when D*A [ 103 (D*B, D*C and D*D), irrespective of the value
of D*A (Fig. 4). The sensitivity of the profile shape to
changes in values of D*A decreases as the difference
between D*A, and the other tracer diffusivities increases.
Thus, the method is not very suitable for the determination
of diffusion coefficients where one of the diffusivities is
much faster (several orders of magnitude) than the rest.
However, this is not a common problem when diffusion of
ions of the same charge on the same lattice site of a given
mineral is considered (see, however, the discussion for Ca
in Part-II).
These aspects have important implications for multicomponent modeling of diffusion profiles in natural garnets
to extract timescales (cooling rates, exhumation rates, or
duration of metamorphism) as well.
Experimental methods
Four new experiments were carried out in Bochum in a
1,000 ton end-loaded piston-cylinder apparatus using WC
cores with 12.7 mm internal diameters and WC pistons and
talc-glass pressure cells with anhydrous parts within a
graphite internal resistance furnace. Parts inside the
graphite furnace of the pressure cell were dried to ensure
that the fO2 was defined very closely by graphite–O2
equilibrium. Generally, the basic experimental approach is
in practice the same as that used in the earlier studies from
the Arizona group (Elphick et al. 1985; Chakraborty and
Ganguly 1992; Ganguly et al. 1998), where a schematic
Contrib Mineral Petrol
0.80
0.80
a
Mg
0.70
Fe
Fe
Fe
Mg
Mg
0.60
N o rm . C o n c .
0.60
N o rm . C o n c .
b
0.70
0.50
0.40
0.30
Mn
0.20
0.50
0.40
0.30
Mn
0.20
Ca
Ca
0.10
0.10
0.00
1
2
3
4
5
6
0.00
7
1
2
Distance [µm]
0.80
4
5
6
0.80
c
Mn
d
Ca
0.70
Fe
Fe
Mg
Mg
0.60
N o rm . C o n c .
0.60
0.50
0.40
0.30
Mn
0.20
0.50
0.40
0.30
Mn
0.20
Ca
Ca
0.10
0.00
7
Distance [µm]
0.70
N o rm . C o n c.
3
0.10
1
2
3
4
5
6
7
Distance [µm]
0.00
1
2
3
4
5
6
7
Distance [µm]
Fig. 4 One set of calculated diffusion profiles (bold lines). Initial
compositions, profile lengths and simulation conditions are equivalent
to experimental run (DPA4) conducted in Ganguly et al. (1998). Fine
lines: maximal adjustment that concentration profiles undergo when
D*-value of one component (in a for Mg, b for Fe, c for Mn, d for Ca)
is increased to approximately three orders of magnitude, while the
other D*-values are held constant, so that D*i *[103 D*j=i. Higher
D*-value of that component would not cause any further change in the
calculated profile shape of any component. Therefore, the gray fields
between bold and fine lines represent the sensitivity range—variation
of profile shapes within these limits would allow a D*-value for any
one of the components to be determined (see case 2 in text).
Sensitivity range (to change of D*-values) of a component becomes
smaller when that component begins to exceed 75 mol. % (see case 1
in text)
cross section of the pressure cell and a more detailed
description of the experimental procedure are available.
Diffusion couples were made of gem-quality single crystals
of natural Fe–Mg garnets where the couple forms a slightly
conic-shaped cylinder with a vertical interface. The
geometry of diffusion couples reduces uniaxial stress during cold compression. Before encasing the highly polished
crystals in the graphite capsule, crystals were cleaned in an
ultrasonic bath successively with acetone, alcohol, soapy
water and deionized water to ensure that the crystals are in
clean contact at the interface. The graphite capsule was
then surrounded by a molybdenum tube of 0.2 mm wall
thickness where the molybdenum tube serves to accomplish two purposes: (1) It reduces the thermal gradient
within the tube in the vertical direction due to excellent
heat conductivity. (2) It ensures that interfaces stay together under hot conditions due to the higher thermal
expansion coefficient of the metal compared to the graphite-garnet assembly. Temperature was measured using a
type-D thermocouple (W 25 %Re–W 3 %Re), and pressure
was calibrated at high temperature using the quartz–coesite
phase transformation. Pressure was increased manually
over a period of 1 h, and then, the samples were annealed
under computer control with a heating rate of 30 °C/min.
After the expiration of experimental duration, the samples
were quenched instantaneously and then decompressed
slowly. To prepare the specimen for electron microprobe
analysis, the diffusion couple was recovered from a high
P–T experiment, sectioned and impregnated with low viscosity epoxy. Each section was carefully oriented within
the epoxy and then polished so that the surface exposed to
electron microprobe analyses was normal to the interface.
Diffusion profiles of all cations present in the natural garnet
samples were measured by step or beam scanning in an
electron microprobe (Cameca SX-50) along a line normal
to the interface. Composition of diffusion couples (see
Fig. 1), experimental run conditions and results from fitting
the diffusion profiles are reported in Table 2.
123
123
Pyr21Alm70Sps4.5Grs4.5- Pyr51Alm37Sps1Grs11
Pyr12Alm77Sps1Grs10- Pyr80Alm20Sps0Grs1
Pyr5Alm73Sps21Grs1- Pyr51Alm37Sps1Grs11
Pyr21Alm70Sps4.5Grs4.5- Pyr51Alm37Sps1Grs11
Pyr28Alm69Sps1Grs2- Pyr47.6Alm46.7Sps1Grs12
Pyr5Alm73Sps21Grs1- Pyr51Alm37Sps1Grs11
Pyr28Alm69Sps1Grs2- Pyr47.6Alm46.7Sps1Grs12
DPA7a
GD13a
DPA5a
DPA10a
R11a
DPA 6a
R17a
38,000
38,000
40,000
38,000
30,000
20,000
26,000
28,000
35,000
1,432
1,375
1,344
1,280
1,250
1,200
1,100
1,057
1,400
1,320
1,300
1,260
T (°C)
94.8
73.5
72.8
218
193
168
576
438
48
120
120
200
t (h)
Ca
842 (386)
361 (178)
281 (106)
88 (31)
195 (66)
113 (29)
15 (7)
10 (5.3)
925 (246)
169 (39)
262 (95)
135 (40)
336 (144)
249 (102)
278 (106)
64 (24)
78 (24)
77 (22)
10 (5)
4 (1.4)
189 (64)
197 (44)
205 (78)
85 (26)
76 (50)
31 (18)
16 (13)
38 (18)
238 (62)
118 (140)
74 (40)
165 (110)
12 (10)
9.5 (5)
26 (18)
0.2 (0.2)
2.5 (2.2)
124 (71)
1,000 (157)
550 (147)
400 (116)
200 (68)
50 (17)
150 (34)
15 (5.3)
10 (4.7)
620 (202)
300 (68)
240 (63)
220 (65)
Mg
Mn
Mg
Fe
D* (V)
D* (NM)
1020 D* (2r) (m2/s)
480 (75)
200 (53)
280 (81)
200 (70)
40 (12)
80 (17)
13 (5.2)
7 (3.9)
540 (176)
190 (43)
150 (30)
95 (28)
Fe
120 (79)
50 (18)
20 (17)
11 (5.1)
270 (69)
Mn
100 (34)
50 (36)
200 (105)
5 (2.8)
20 (12)
20 (9.1)
0.8 (0.6)
0.5 (0.4)
150 (34)
Ca
0.84
0.66
0.70
0.44
3.90
0.75
1.00
1.03
1.50
0.56
1.09
0.62
Mg
0.70
1.25
1.00
0.32
1.95
0.97
0.73
0.57
0.35
1.03
1.36
0.9
Fe
0.63
0.61
0.78
3.45
0.88
Mn
D* (NM)/D* (V)
1.18
1.48
0.82
2.49
0.47
1.29
0.24
4.92
0.82
Ca
a
Results from our adjusted model fits to experimental data reported in Ganguly et al. (1998)
Two sets of diffusion coefficients are shown—those fitted by using the Nelder-Mead (NM) algorithm and ‘‘visually’’ (V), respectively. All results are corrected for the convolution effect of the microprobe beam. ±2r
errors for diffusion coefficients fitted by (V)-method are estimated using the accuracy of the distance measurement. Error of (NM) method additionally considers the error arising from the scatter of the data
Pyr43Alm44Sps2Grs11- Pyr48Alm39Sps2Grs11
Pyr5Alm73Sps21Grs1- Pyr51Alm37Sps1Grs11
DPA4a
25,000
Pyr53Alm33Sps1Grs9Adr3Pyr74Alm13Sps1Grs4Adr3Uv4
SMG6
SMG0
25,000
Pyr52Alm39Sps4Grs4Adr1Pyr73Alm15Sps1Grs5Adr3Uvr4
SMG1
25,000
Pyr60Alm28Sps1Grs11Pyr74Alm14Sps1Grs5Adr3Uvr4
SMG5
P (bar)
Diffusion couple
Run no.
Table 2 Summary of annealing conditions and self-diffusion data of divalent cations in pyrope–almandine diffusion couples
Contrib Mineral Petrol
Contrib Mineral Petrol
Determination of diffusion coefficients
Diffusion coefficients were retrieved from the measured
concentration profiles using two different approaches. As a
first approach, the minimization of the L1-distance between
the measured and the computed profile according to the
Nelder-Mead algorithm outlined above was used. The
‘‘visual fitting’’ approach of Loomis et al. (1985) was used
to carry out forward modeling numerically, where diffusion
profiles were calculated for chosen values of the four tracer
diffusion coefficients (D*Fe, D*Mg, D*Mn and D*Ca), convolved
using the approach outlined above and compared to the
measured profiles (Fig. 5). The chosen diffusion coefficients were varied until the best description of the
measured profile shapes was obtained. The input D* values
used to obtain such a profile were considered to be the
relevant tracer diffusion coefficients at that particular run
condition. Diffusion coefficients retrieved according to
both methods are reported in Table 2. The statistical errors
for the tabulated values of self-diffusion coefficients
derived from the electron microprobe data have been calculated as follows, taking into account the scatter of the
data and resolution of step scan and/or beam scan profiling,
which produces an uncertainty of length of the diffusion
profile due to a stretching or contraction of the distance
scale (&0.5 lm). Thus, the variance of a D* value, r2*D,
including these errors, may be approximated as (Sano et al.
2011):
rD r2D ðsÞ þ r2D ðxÞ
where r2*D(s) and r2*D(x) are, respectively, the variance of D*
arising from the two sources of errors described above. The
first term considers the scatter of the data and has been
estimated from the error of the optimization procedure by
Nelder-Mead only. This term has not been taken into
consideration for ‘‘visual fitting’’ of profiles. Following
Sano et al. (2011), the second term caused by the
uncertainty of diffusion distance is approximated as
"
#
ldp þ rx 2
rD ðxÞ D
1
ð10Þ
ldp
a
Interface
Diffusion
profile
Prp52
Prp73
50.00 µm
0.80
BSE
15.0 kV
500
b
0.70
Mg
N o rm . C o n c.
0.60
0.50
0.40
Fe
0.30
0.20
Ca
0.10
Mn
0.00
1
2
3
4
5
ð9Þ
6
7
Distance [µm]
Fig. 5 a Back-scattered electron image of a diffusion couple (run
SMG 1) annealed at 25 kbar, 1,300 °C for 120 h in a graphite
capsule. Dark region represents the pyrope richer garnet crystal. Line
across the interface represents the diffusion profile measured along a
crack free region. b Corresponding Mg, Fe, Mn and Ca diffusion
profiles. Lines through data points represent convolution corrected
simulations, which were fitted by the following set of diffusion
coefficients (Mg 2.62 9 10-18, Fe 2.05 9 10-18, Mn 2.38 9 10-18,
Ca 1.24 9 10-18 m2/s)
where ldp is the best estimate of the total diffusion distance
and rx the estimated error of ldp. Note that other important
sources of errors that may be present (e.g., from the
reproducibility of experimental run conditions, or the use
of garnets of somewhat different compositions in the diffusion couple experiments) are not included in these
values.
It is found that the D*Mg [ D*Fe at any given run condition, which is consistent with the findings in Fe–Mg systems in earlier studies (Elphick et al. 1985; Ganguly et al.
1998; Perchuk et al. 2009). D*Mn is poorly constrained
(see also the discussion above), and D*Ca requires special
handling—this is discussed in the companion paper.
In order to obtain the best possible constraints on
Arrhenius parameters, we have used data from diffusion
profiles from all experiments carried out with Fe–Mg diffusion couples in piston-cylinder apparatus using a similar
setup. The profiles were refitted using the improved techniques developed in the paper. We have shown that both
techniques, (a) Nelder-Mead optimization and (b) ‘‘visual
fitting’’, yield reliable diffusion coefficients and differ for
Mg and Fe by a factor of *3 at most (Table 2). However,
in order to have a set of most reliable diffusion coefficients,
we have used the first approach for our final data analysis
(Fig. 6).
123
Contrib Mineral Petrol
1400
1200
1300
term in the above expression represents an effective
activation volume that incorporates the pressure effect of
variation of fO2 along the reaction curve as well (Ganguly
et al. 1998; Holzapfel et al. 2007). Here, we have assumed,
based on results from earlier studies (e.g., Ganguly et al.
1998), that the activation volume of diffusion is not
strongly variable for different garnet compositions and so
we have taken the values of DV? from Chakraborty and
Ganguly (1992) to be applicable to the Fe–Mg garnets as
-1
well. These are DV?
mol-1, DV?
Mg = 0.53 J bar
Fe = 0.56
-1
-1
J bar mol . With these activation volumes, fits to the
diffusion coefficients retrieved from the experimental data
yield the following values for the parameters in Eq. (11):
1100 °C
-16.5
-17.0
Mg
2
l o g D ( m /s )
-17.5
-18.0
-18.5
Mg: Q1bar = 228.3 ± 20.3 kJ/mol,
D0 = (2.72 ± 4.52) 9 10-10 m2/s,
Fe: Q1bar = 226.9 ± 18.6 kJ/mol,
D0 = (1.64 ± 2.54) 9 10-10 m2/s.
-19.0
Fe
-19.5
6.00
6.25
6.50
6.75
7.00
7.25
7.50
4
10 /T(K)
Fig. 6 Arrhenius plot of experimentally determined self-diffusion
coefficients of Mg (diamonds) and Fe (squares) in garnet fitted by the
‘‘visual’’ method. The error bars on diffusion data indicate approximately ±2r estimates (95 % confidence interval) of the individual
D*-values, when only errors of measurement of composition and
distance are considered (see text for more details). Solid lines are
resulting from weighted least squares fits, whereas dotted lines
represent the 2r envelopes of the log D*-values predicted from the
Arrhenius relations and 2r errors of the Arrhenius parameters. All
data have been normalized to a pressure of 25 kbar and fO2
corresponding to those defined by the presence of graphite in the
system C–O–H. Filled symbols: diffusion data determined as part of
this experimental study, open symbols: diffusion data determined in
this study by refitting the experimental profiles of Ganguly et al.
(1998)
Given the limited number of data points that are available even after the repeated efforts at measuring these
experimentally demanding diffusion coefficients, it is
prudent to include as many data sets as possible in order to
get meaningful Arrhenius parameters. For older data, it is
also indicated where the data were reported for the first
time and when the diffusion coefficients obtained in this
study by refitting the earlier profiles are different from
the values reported in the original papers. Assuming the
activation volume, DV?, to be independent of pressure, the
diffusion coefficient of a species as a function of pressure
and temperature is given by
Q1bar þ DV þ ðP 1Þ
2:303RT
QP
:
¼ log D0 2:303RT
log D ðP; TÞ ¼ log D0 ð11Þ
The diffusion coefficients in this study are determined
from experiments at fO2 conditions along a curve defined
by graphite–oxygen–hydrogen equilibrium in the pressure
cell (imposed by the graphite capsule). Therefore, the DV?
123
Arrhenius parameters were calculated using the method
of least squares, and Arrhenius slopes for Mg and Fe are
shown in Fig. 6. Their ±2r error envelopes were determined according to Tirone et al. (2005) as
2
1
2
2
rlog D ðTÞ ¼ rlog D0 þ
r
T QP=2:303R
2
QP
þ cov log D0 ;
ð12Þ
T
2:303R
where r is the standard deviation, cov(logD0, Qp/2.303R) is
the covariance of intercept (logD0) and slope (Qp/
2.303R) (e.g., see Tirone et al. 2005). Uncertainties of
results presented here contain only errors from the sources
described above. The new results, obtained using a different population of natural garnets from the ones used in
the study of Ganguly et al. (1998), demonstrate that (a) for
the given range of Fe–Mg solid solution, the relatively
more Mg-rich diffusion couples spanning a smaller compositional range (and hence more amenable to visual as
well as numerical fitting) yield comparable diffusion
coefficients, (b) differences in minor and trace elements
between garnets do not have a significant effect on diffusion rates of elements such as Fe and Mg.
Discussion
Diffusion coefficients (D*Mg and D*Fe) obtained in this study
are compared to results from earlier studies in Fig. 7.
The consequences for modeling processes in metamorphic garnets using the results from this study may be
explored by extreme extrapolation from the experimental
run conditions to a P and T of 5 kbar and 600 °C, respectively. Tracer diffusion coefficients that would be obtained
using the expressions given above can be compared to
Contrib Mineral Petrol
a
(Cygan and Lasaga 1985), 3.0(Chakraborty and Rubie
1996), 242 (Schwandt et al. 1995), 2.1 (Ganguly et al.
1998), 1.5 (Carlson 2006) and 10.4 (Perchuk et al. 2009).
For Fe tracer diffusion, h = 32 (Ganguly et al. 1998),
2.6 (Carlson 2006) and 1.9 (Perchuk et al. 2009). At this
pressure condition, results from this study indicate that
D*Mg/D*Fe lie within a factor of *1.4 between 600 and
1,400 °C, or in other words, the results obtained from
additional experiments and improved fits in this study lie
within a factor of 1.0–3.0 of earlier results, with the exceptions of D*Mg from Schwandt et al. (1995), D*Fe from Ganguly
et al. (1998) and D*Mg from Perchuk et al. (2009). 1/h of the
D*Mg is the approximate factor in each case by which timescales retrieved using the earlier results would differ from
those that would be obtained using the present data set.
Implications for resetting and closure of garnets
in natural settings
b
Fig. 7 Arrhenius plot of (a) Mg and (b) Fe tracer/tracer diffusion
coefficients determined in this study (solid bold lines) with earlier
data. All data have been normalized to a pressure of 25 kbar and fO2
corresponding to those defined by graphite in the system C–O–H.
Abbreviations: C06 Carlson (2006) obtained from earlier experimental data and modeling result of natural stranded diffusion profiles,
CG92 (Chakraborty and Ganguly 1992) & G98 (Ganguly et al. 1998).
D*-values retrieved from experimental data in Alm-Sps and Alm-Prp
diffusion couples, respectively; CL85 (Cygan and Lasaga 1985),
CR96 (Chakraborty and Rubie 1996) & S95 (Schwandt et al. 1995).
D*(Mg) in natural Prp garnets, L85 (Loomis et al. 1985) (these data
were incorporated in G98), P09 (Perchuk et al. 2009)
Diffusion data in garnet find application in several situations. In addition to the modeling of frozen compositional
profiles for geospeedometry, two common and related
questions are: (i) For what kind of thermal histories can
garnets of a given size retain their chemical compositions
at the core? (ii) During cooling, at what temperatures do
compositions at the cores of garnets of a given size freeze
in? These are issues related to the concept of a closure
temperature that are usually evaluated using the formulation developed by Dodson (1973). This formulation is,
however, based on certain simplifying assumptions. One of
these is that the compositional zoning in the mineral of
interest should not retain any memory of the initial concentration distribution, that is, even at the core of the
crystal the initial concentration should be reset by diffusion. This is why there is no term accounting for the initial
concentration distribution or peak temperature in the formulations of Dodson. In garnets, however, this is frequently not the case; more often than not, diffusional
zoning does not penetrate to the core.
For such situations, Ganguly and Tirone (1999) have
developed a formulation where they showed that the core
composition of a spherical grain surrounded by a homogeneous infinite matrix (e.g., garnet surrounded by a large
mass of biotite or a matrix with fast grain boundary diffusion) is not affected if the value of a dimensionless
variable, M, is B0.1 for thermal histories with peak temperature, T0 B 1,000 °C. The parameter M is defined as
M¼
results using expressions given in other studies on Fe–Mg
garnets in terms of a factor h, where h[D*i from study
X] = [D*i from this study]. For Mg tracer diffusion, h = 1.0
RDðT0 ÞT 2
QðdT=dtÞa2
ð13Þ
where D(T0) is the diffusion coefficient at the peak temperature, a is a characteristic grain dimension (radius for a
123
Contrib Mineral Petrol
sphere and infinite cylinder and half-thickness for an infinite plane sheet) and dT/dt is the cooling rate at T (the
temperature that appears in the numerator). The cooling
curve is assumed to follow an asymptotic relation in which
the reciprocal temperature varies linearly with time
(1/T = 1/T0 ? gt) (it is formally the same as that used by
Dodson (1973) in his derivation of closure temperature
formulation). Using Eq. (13), Tirone and Ganguly (2010)
addressed the problem of resetting of core composition of
garnet grains as a function of peak temperature, T0, and
cooling rate. Following their approach, we illustrate
(Fig. 8, similar to Fig. 2 of Tirone and Ganguly 2010)
minimum grain sizes for which garnets would retain their
core compositions (for a binary chemical diffusion coefficient, D(Fe–Mg) at XFe = 0.75 and XMg = 0.25, which is
taken to be a typical composition for metapelitic garnet).
The D(Fe–Mg) values can be calculated from Eq. 8, using the
D*Fe and D*Mg values from this work.
The results in Fig. 8 show that the minimum grain size
to preserve core composition is revised upwards compared
to those reported in Tirone and Ganguly (2010) by a factor
of 2.8 at 700 °C, 2.2 at 800 °C and 1.8 at 900 °C, independent of cooling rate. This difference arises mainly
because the values of D*Fe that are calculated at these
temperatures using data from this work are higher than
those calculated using Ganguly et al. (1998).
6.00
Ganguly and Tirone (1999) calculated closure temperature profiles in crystals surrounded by a homogeneous
infinite matrix for different values of T0, M and crystal
geometry. The results show that for T0 in the range of
700–1,100 °C, the temperatures calculated from the rim
and matrix compositions are significantly lower than T0 if
M [ 0.01. Using this value of M and the diffusion data
presented in this paper, we have calculated the size of
garnet grains below which the rim compositions would
reflect significant resetting of temperature during cooling if
these are surrounded by a homogeneous infinite matrix.
The results are illustrated in Fig. 9.
The basic observations from Figs. 8 and 9 are that
(i) slow cooling (*1–2 °C/my) from temperatures[750 °C
would reset the cores of several-mm-sized garnet grains
that are enclosed in a medium where diffusion rates are
infinitely fast (e.g., an infinite mass of biotite, rather than
pyroxenes), (ii) rapid cooling ([100 °C/my) would almost
always retain compositions from metamorphic peak at
cores of crystals. On the other hand, it would be almost
impossible to not reset the rim compositions of garnets
cooling from temperatures[750 °C unless cooling is faster
than several 1,000 °C/my (Fig. 9). For medium-grade
metamorphic rocks (500–600 °C), the rim compositions
are likely to retain (Fig. 9) the peak metamorphic compositions unless cooling is exceedingly slow. Thus, these
diagrams may be used to estimate whether rim or
core compositions should be used to determine peak
6.00
5.00
5.00
r
/My
5
10
20
3.00
20
20
10
2.00
100
10
0
10
1.00
2 °C
5
2
5
2.00
Radius (mm)
4.00
3.00
2°
C/M
yr
R a d iu s ( m m )
4.00
100
1.00
0.00
650
700
750
800
850
T0 (°C)
0.00
Fig. 8 Radius of spherical grains that would retain their core
compositions without resetting, for cooling from different peak
temperatures, at different rates. The calculations are shown for binary
chemical diffusion in a garnet of composition XFe = 0.75,
XMg = 0.25. Bold solid lines show results obtained using diffusion
coefficients reported in this work; fine solid lines show the results
obtained by Tirone and Ganguly (2010) using data from Ganguly
et al. (1998) for comparison
123
650
700
750
800
850
T0 (°C)
Fig. 9 Curves showing grain sizes for which there would be
significant resetting of rim compositions, for different peak temperatures and cooling rates. Calculations are for a chemical diffusion
coefficient in a binary garnet of compositions XFe = 0.75,
XMg = 0.25 (same as in Fig. 8)
Contrib Mineral Petrol
metamorphic temperatures at different grades. Curves such
as the ones illustrated in Figs. 8 or 9 can be easily calculated using the Arrhenius parameters presented here and
Eq. (10) for any arbitrary garnet size and cooling rate.
Conclusions
We have used numerical simulations and data from new
experiments to clarify several aspects of multicomponent
diffusion in garnets. Comparison of the compositions of
experimental garnet diffusion couples and those commonly
found in nature in metamorphic and mantle-derived rocks
indicates that the natural garnet compositions, with the
exception of some rather Ca-rich garnets from eclogites,
are well represented by the experimental couples. Diffusion
in Mn-rich, low-to-medium-grade pelitic garnets occur
along different compositional vectors and have a different
behavior (D*Fe*D*Mg). Therefore, these should be modeled
using different diffusion coefficients from those used for
modeling diffusion in more Mn-poor, Fe–Mg garnets
(D*Fe \ D*Mg). In this study, we have focused on the diffusion behavior in garnets of the second kind.
Incorporation of the effects of thermodynamic nonideality in the diffusion models alters the shapes of diffusion profiles. Therefore, use of non-ideal models helps to
obtain better numerical fits to measured concentration
profiles. Calculated diffusion penetration distances (or
equivalently, retrieved diffusion coefficients from a given
profile length) differ significantly between ideal and nonideal models for some unusual Mg–Mn–Ca-rich garnets
only. Accounting for these effects, plus a more generalized
convolution correction to profile shapes, allows us to obtain
better defined diffusion coefficients from experimental
runs.
We have developed and tested a numerical method to
calculate best-fit diffusion coefficients from experimentally
induced concentration profiles. This method yields diffusion coefficients that are similar to the visual fitting method
that has been used until now. The latter has the advantage
that a larger range of diffusion couple compositions can be
analyzed using the method because effects of compositional dependence of diffusion coefficients and asymmetry
of profile shapes, which occur when the difference in
composition between the two ends of the diffusion couple
is large, can be handled. Here, we present data obtained
using the objective numerical fitting method.
The numerical calculations, combined with a theoretical
analysis, further revealed that there are limitations to the
application of these methods to extract multiple diffusion
coefficients from a single run. When one of the diffusion
coefficients becomes much faster or slower than the rest, or
when the diffusion couple has a composition that is
dominated by one component ([75 %), then profile shapes
become insensitive to one or more diffusion coefficients
and these cannot be retrieved by modeling the concentration profiles.
We have carried out four new diffusion couple experiments at pressures of 25 kbar and at temperatures between
1,260 and 1,400 °C. Results from these experiments were
combined with earlier results obtained using a similar
experimental setup to obtain better constrained Arrhenius
parameters for diffusion in Fe–Mg garnets. Consistency of
results obtained in the new experiments that used a different population of garnets with the older results indicates
that (a) for primarily Fe–Mg solid solutions, it is possible to
use a single set of Arrhenius parameters to describe the
diffusion behavior reasonably well and (b) trace element
compositions of garnets do not affect the diffusion coefficients of Fe and Mg substantially. It is found that
D*Fe \ D*Mg not only at experimental conditions, but also on
extrapolation down to lower temperatures. The diffusion
rates obtained using these parameters indicate that (a) it
may be difficult to preserve UHT conditions in cores of
garnets that are enclosed in a matrix where diffusion is
infinitely fast, (b) rims of high-grade garnets will always be
reset and (c) rims of low-to-medium-grade garnets will
record peak temperature conditions at their rims.
Acknowledgments We thank the German Science Foundation
(DFG) for generously supporting this work. Thanks are due to J. Van
Orman and an anonymous reviewer for constructive reviews. SAB
was supported by the SFB 526 Program of the German Science
Foundation and an INSA-DFG Fellowship funded the visit of SKB to
Bochum. JG gratefully acknowledges the support from Alexander
Humboldt foundation revisit program and US National Science
Foundation grant No. EAR-1016189 for his participation in this
project.
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